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\title{《基础复分析》第10章解析延拓 - 习题}
\author{CGZ ET AL}

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%## 《基础复分析》习题十

\begin{enumerate}

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\item % 1

设幂级数 $f(z) = \sum a_n z^n$ 的收敛半径为 1, 且 $a_n \geq 0$. 试证 $\zeta = 1$ 是 $f(z)$ 的奇点.
(提示: 将 $f(z)$ 展开为 $z-1/2$ 的幂级数.)


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\item % 2

试证函数
$$
f(z) = \sum_{n=0}^{\infty} z^{2^n} = z + z^2 + z^4 + z^8 + \cdots
$$
在单位圆周上没有正则点.


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\item % 3

设 $(f,D), (g,D)$ 是两个解析函数芽, $P(\cdot,\cdot)$ 是两个变量的多项式, 且在 $D$ 内 $P(f,g)=0$. 如果 $(f,D), (g,D)$ 能沿一条曲线分别解析延拓到 $(f_1,D_1)$ 和 $(g_1,D_1)$, 试证在 $D_1$ 内 $P(f_1,g_1)=0$.


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\item % 4

设 $f(z)$ 在区域 $\Omega$ 上解析, 且存在一点 $a \in \Omega$, 使得
$$
\lim_{z \to \partial \Omega} |f(z)| > |f(a)|.
$$
试证如果 $a$ 是 $f(z)-f(a)$ 的 $k$ 阶零点, 则 $f(z)$ 在 $\Omega$ 内至少有 $k$ 个零点 (计重数).






\end{enumerate}

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